I am an ex-RAF QFI so, if I can think that far back, here goes:
Instrument Rating Exam Testers are Picked - IRET/PICD.....
IAS - what you see in the cockpit
Rectified AS - IAS corrected for Pressure and Instrument error
Equivalent AS - RAS corrected for Compressibility
TAS - EAS corrected for Density error
Technically, EAS is what you use for stall speed, approach speed, aerobatic speeds, lift, drag etc. It affects how the aircraft "feels" and "flies". It is a measure of the number of molecules going past the airframe and over the wing. So, in that respect, it is what we use for aircraft performance. Since we don't have air data computers to make the necessary corrections then we use IAS which is close enough for our needs.
As altitude increases, air density reduces. So to get the same IAS (number of molecules passing the aircraft), we have to fly faster - this is the so-called density error - that is, our true speed through the air is greater than indicated on the instrument. When looking at range and fuel consumption in still air, this is the number you use. It also affects TO and landing performance in that at high altitude/temperature, you have to accelerate to a higher TAS to get the same IAS for the necessary lift to take-off (plus the engine produces less power so there's a double whammy). Likewise on landing, for a given IAS, your TAS is higher and so the landing roll is longer.
Ground Speed is totally irrelevant to the aerodynamic performance of the aircraft. It is the TAS corrected for wind. It is also what is given by GPS. In that respect, it is what affects navigation performance but unless one is trying to get a Time on Target or similar, one does not "fly" GS. Other than changing altitude to get a more favourable wind, you cannot "fly" a particular GS to improve performance. You get what you get and use that to calculate headings and times.
So, you set your power which gives you an IAS. Using your whizz-wheel, you then calculate your TAS for your altitude/temperature. Using this number, you apply the wind and calculate drift and GS. Using these numbers, you then know your required heading to make good your track and the time to cover the leg distance. With and EFIS and GPS, this is all done for you.
As for the TAS Vne: Vne (in the VANS aircraft at least) is not due to a structural airframe issue, it is do to control surface flutter considerations. Since air density and therefore aerodynamic damping reduce with altitude, the IAS needs to be reduced - hence the TAS limit rather than IAS.